Ahlfors Theorems for Differential Forms

and Applied Analysis 3 Let y1, . . . , yk be an orthonormal system of coordinates in R, 1 ≤ k ≤ n. Let A be a domain in R, and let B be an n − k -dimensional Riemannian manifold. We consider the manifold N A × B. 2. Boundary Sets Below we introduce the notions of parabolic and hyperbolic type of boundary sets on noncompact Riemannian manifolds and study exhaustion functions of such sets. We also present some illuminating examples. Let M be an n-dimensional noncompact Riemannian manifold without boundary. Boundary sets on M are analogies to prime ends due to Carathéodory cf. e.g., 2 . Let {Uk}, k 1, 2, . . . be a collection of open setsUk ⊂ Mwith the following properties: i for all k 1, 2, . . ., Uk 1 ⊂ Uk, ii ⋂∞ k 1 Uk ∅. A sequence with these properties will be called a chain on the manifoldM. Let {U′ k }, {U′′ k} be two chains of open sets on M. We will say that the chain Uk is contained in the chain {U′′ k }, if for each m ≥ 1 there exists a number k m such that for all k > k m we have U′ k ⊂ U′′ m. Two chains, each of which is contained in the other one, are called equivalent. Each equivalence class ξ of chains is called a boundary set of the manifoldM. To define ξ it is enough to determine at least one representative in the equivalence class. If the boundary set ξ is defined by the chain {Uk}, then we will write ξ {Uk}. A sequence of points mk ∈ M converges to ξ if for some and, therefore, all chain {Uk} ∈ ξ the following condition is satisfied: for every k 1, 2, . . . there exists an integer n k such that mn ∈ Uk for all n > n k . A sequence mn lies off a boundary set ξ {Uk}, if for every k 1, 2, . . . there exists a number n k such that for all n > n k mn /∈Uk. A boundary set ξ {Uk} is called a set of ends of the manifoldM if each of {Uk} has a compact boundary ∂Uk. If in addition each of the setsUk is connected, then ξ {Uk} is called an end of the manifoldM. 2.1. Types of Boundary Set Let D be an open set on M and let A,B ⊂ D be closed subsets in D such that A ∩ B ∅. Each triple A,B;D is called a condenser on M. We fix p ≥ 1. The p-capacity of the condenser A,B;D is defined by